Problem: Solve for $x$ and $y$ by deriving an expression for $y$ from the second equation, and substituting it back into the first equation. $\begin{align*}-7x+4y &= -8 \\ 5x-6y &= 1\end{align*}$
Answer: Begin by moving the $x$ -term in the second equation to the right side of the equation. $-6y = -5x+1$ Divide both sides by $-6$ to isolate $y$ $y = {\dfrac{5}{6}x - \dfrac{1}{6}}$ Substitute this expression for $y$ in the first equation. $-7x+4({\dfrac{5}{6}x - \dfrac{1}{6}}) = -8$ $-7x + \dfrac{10}{3}x - \dfrac{2}{3} = -8$ Simplify by combining terms, then solve for $x$ $-\dfrac{11}{3}x - \dfrac{2}{3} = -8$ $-\dfrac{11}{3}x = -\dfrac{22}{3}$ $x = 2$ Substitute $2$ for $x$ back into the top equation. $-7( 2)+4y = -8$ $-14+4y = -8$ $4y = 6$ $y = \dfrac{3}{2}$ The solution is $\enspace x = 2, \enspace y = \dfrac{3}{2}$.